Statement-I :
If `(x^(2)+3x+1)/(x^(2)+2x+1)=A+(B)/(x+1)+(C)/((x+1)^(2))`, then `A+B+C=0`
Statement-II:
If `(x^(2)+2x+3)/(x^(3))=A/x + (B)/(x^(2))+(C)/(x^(3))`, then `A+B-C=0`
Which of the above statements is true

Options

A. only I is true
B. only II is true
C. both I & II are true
D. neither I nor II true

Answer

To solve the problem, we need to analyze both statements separately and verify their validity.

### Statement I:
We have the expression:
\[ \frac{x^2 + 3x + 1}{x^2 + 2x + 1} = A + \frac{B}{x+1} + \frac{C}{(x+1)^2} \]

1. **Factor the denominator**:
The denominator \(x^2 + 2x + 1\) can be factored as:
\[ (x + 1)^2 \]
So we rewrite the equation:
\[ \frac{x^2 + 3x + 1}{(x + 1)^2} = A + \frac{B}{x + 1} + \frac{C}{(x + 1)^2} \]

2. **Combine the right-hand side**:
To combine the right-hand side, we need a common denominator:
\[ A + \frac{B}{x + 1} + \frac{C}{(x + 1)^2} = \frac{A(x + 1)^2 + B(x + 1) + C}{(x + 1)^2} \]

3. **Set the numerators equal**:
Now, we equate the numerators:
\[ x^2 + 3x + 1 = A(x + 1)^2 + B(x + 1) + C \]

4. **Expand the right-hand side**:
Expanding \(A(x + 1)^2\):
\[ A(x^2 + 2x + 1) = Ax^2 + 2Ax + A \]
Expanding \(B(x + 1)\):
\[ Bx + B \]
Thus, the right-hand side becomes:
\[ Ax^2 + (2A + B)x + (A + B + C) \]

5. **Compare coefficients**:
From the left-hand side \(x^2 + 3x + 1\), we have:
- Coefficient of \(x^2\): \(A = 1\)
- Coefficient of \(x\): \(2A + B = 3\)
- Constant term: \(A + B + C = 1\)

6. **Substituting \(A\)**:
Since \(A = 1\):
- From \(2A + B = 3\):
\[ 2(1) + B = 3 \implies B = 1 \]
- From \(A + B + C = 1\):
\[ 1 + 1 + C = 1 \implies C = -1 \]

7. **Calculate \(A + B + C\)**:
\[ A + B + C = 1 + 1 - 1 = 1 \]
Thus, Statement I is **false** because \(A + B + C \neq 0\).

### Statement II:
We have the expression:
\[ \frac{x^2 + 2x + 3}{x^3} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} \]

1. **Combine the right-hand side**:
The right-hand side can be written as:
\[ \frac{A x^2 + B x + C}{x^3} \]

2. **Set the numerators equal**:
Equating the numerators gives:
\[ x^2 + 2x + 3 = A x^2 + B x + C \]

3. **Compare coefficients**:
From the left-hand side \(x^2 + 2x + 3\), we have:
- Coefficient of \(x^2\): \(A = 1\)
- Coefficient of \(x\): \(B = 2\)
- Constant term: \(C = 3\)

4. **Calculate \(A + B - C\)**:
\[ A + B - C = 1 + 2 - 3 = 0 \]
Thus, Statement II is **true**.

### Conclusion:
- Statement I is false.
- Statement II is true.

### Final Answer:
Only Statement II is true.